© Fluent Inc. 11/29/2015E1 Fluent Software Training TRN-98-006 Using the Segregated and Coupled...

© Fluent Inc. 04/18/23E1

Fluent Software TrainingTRN-98-006

Using the Segregated and Coupled Solvers



Outline Introduction Solution procedure Terminology Convergence

What is convergence? Judging and monitoring convergence Accelerating convergence

Stability Accuracy

Discretization schemes Grid adaption Grid independence



FLUENT 5 provides two classes of numerical methods Segregated solver (FLUENT/UNS, FLUENT 4.5) Coupled solver

Explicit (RAMPANT) Implicit

Can switch “on-the-fly” between these methods Solution procedure that you follow is basically the same for all

solvers. Both classes of methods employ similar discretization schemes (finite-

volume) but different linearization and solution methods

Introduction



Linearization: Implicit vs. Explicit

Discrete nonlinear governing equations need to be linearized so that system of equations can be solved for dependent variables in the cells

Implicit linearization - unknown value in each cell uses existing and unknown values from neighboring cells

System of equations obtained which must be solved simultaneously In segregated approach, each variable equation linearized implicitly with

respect to itself; variable field solved over all cells In coupled approach, each equation in coupled set linearized implicitly

with respect to all dependent variables; block coupled equations solved Explicit linearization - unknown value in each cell computed from

relations that include only existing values Each equation in coupled set linearized explicitly Multi-stage (Runge-Kutta) solver can be used



Solution Procedure: What You Do

Modify solution parameters or grid

NoYes

NoYes

Set the solution parameters

Initialize the solution

Enable the solution monitors of interest

Calculate a solution

Check for convergence

Check for accuracy

Stop



Segregated Solution Procedure: What the Solver Does

Update properties.

Solve momentum equations (u, v, w velocity).

Solve pressure-correction (continuity) equation. Update pressure, face mass flow rate.

Solve energy, species, turbulence, and other scalar equations.

Converged?

StopNo Yes



Coupled Solution Procedure:What the Solver Does

Update properties.

Solve continuity, momentum, energy, and species equations simultaneously.

Converged?

StopNo Yes

Solve turbulence and other scalar equations.



Unsteady Solution Procedure Same procedure for segregated and time-implicit coupled solvers:

Execute segregated or coupled procedure, iterating to convergence

Take a time step

Requested time steps completed?

No Yes Stop

Update solution values with converged values at current time

Time-explicit coupled solver explicitly solves for all variables one cell at a time at each time step (Runge-Kutta).



What is Convergence? At convergence:

All discrete conservation equations (momentum, energy, etc.) are obeyed in all cells to a specified tolerance.

Solution no longer changes with more iterations. Overall mass, momentum, energy, and scalar balances are obtained.

“Residuals” measure imbalance (or error) in conservation equations. The governing transport equations are solved iteratively. Final solution (ideal): Iteration i:

Opposite of convergence is divergence. Imbalance grows larger with more iterations.

Note: Convergence is not the same as accuracy!

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nbnbpp Rbaanb

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Judging Qualitative Convergence

Generally, a decrease in residuals by 3 orders of magnitude (tolerance=110-3 ) indicates at least qualitative convergence.

Major flow features established Exceptions:

In segregated solver scaled energy residual must decrease to 110-6. Scaled species residual may need to decrease to 110-5 to achieve species

balance. Monitor other variables and balances to ensure quantitative

convergence.

Beware: Unconverged results can be misleading!



Residual Plots

Residual plots show: When the residual values have reached the specified tolerance Which equations are having convergence trouble

all equations converged

continuity equation convergencetrouble affects convergence ofall equations



Monitoring Quantitative Convergence

In addition to residuals, you can also monitor:

Lift, drag, or moment

Solve Monitors Force... Value of a variable or function:

At a boundary On a defined surface At a point

Solve Monitors Surface...

When the monitored value stops changing, the solution is converged.

Monitoring average static temperature over a surface



Convergence Monitor Plots

When lift value levels off, solution is converged.



Checking Balances

In addition to monitoring residual and variable histories, you should also check for overall heat and mass balances.

Net imbalance should be less than 0.1% of net flux through domain.

Report Fluxes...



Decreasing the Convergence Tolerance

Recall: at convergence, all discrete conservation equations are obeyed in all cells to a specified tolerance.

If your residuals indicate that the solution is converged, but the solution is still changing or has a large mass/heat imbalance:

Try reducing the convergence tolerance.

Solve Monitors Residual... Then calculate until solution converges to the new tolerance.



Accelerating Convergence

Convergence can be accelerated by: Supplying good initial conditions Modifying under-relaxation parameters or changing the Courant number

(coupled solver) Controlling multigrid solver settings



Supplying Initial Conditions

Initialize all solution variables before calculating a solution.

Solve Initialize Initialize... Also improves solution stability In some cases, correct initial guess is required:

Example: nozzle will not attain supersonic flow unless initial flow condition is supersonic.

“Patch” values for individual variables in certain regions.

Solve Initialize Patch... Free jet flows (patch high velocity for jet) Combustion problems (patch high temperature for ignition)

Start from a previous solution. Often used in conjunction with patching.



Starting from a Previous Solution

Previous solution can be used as an initial condition when changes are made to problem definition.



What is Under-relaxation? Equation set being solved is non-linear. Equation for one variable may depend on other variables, e.g.,

Temperature Mass fraction

For solver stability, the change in a variable’s value between iterations is reduced by an “under-relaxation” factor, :

e.g., under-relaxation of 0.2 limits the change in P to 20% of the computed change of for one iteration.

Under-relaxation factors for equations outside coupled set (for coupled solver) are set just as in segregated solver

P P P ,old



Modifying Under-relaxation Factors

Use default under-relaxation parameters to start a calculation.

To accelerate convergence, increase under-relaxation factor for an equation if its residual is either:

Monotonically decreasing Oscillating very slightly, but decreasing

Solve Controls Solution...

Increasing under-relaxation factors can also create instability! Leads to divergence! Good idea to save intermediate data before increasing under-relaxation.



Residual plot showing good convergence behavior accelerated by an increase in under-relaxation:

Example: Accelerating Convergence

Under-relaxation increased here



The Multigrid Solver

“Multigrid” solver accelerates convergence for: Large number of cells Large cell aspect ratios

x/y > 20 or so, for example Large differences in thermal conductivity

Such as in conjugate heat transfer General concept of multigrid is the same for structured and

unstructured grids, although the implementation is different.



The Multigrid Concept (1) Multigrid solver uses sequence of grids going from fine to coarse.

You choose the number of coarse grid levels. Solver creates them automatically.

Influence of boundaries and far-away points more easily transmitted to interior on coarse meshes than on fine meshes.

In coarse meshes, grid points closer together in computational space. Fewer computational cells between any two spatial locations.

However, fine meshes give more accurate solutions.

original grid coarse grid level 2

coarse grid level 1



The Multigrid Concept (2) Use solution on coarse meshes as starting point for solution on finer meshes.

Coarse-mesh solution contains influence of boundaries and far neighbors. These effects felt more easily on coarse mesh.

Accelerates convergence on fine mesh

Final solution obtained for mesh you created. Coarse mesh calculations:

Only speed up convergence Do not change final answer

fine mesh

corrections

summed equations (or volume-averaged solution)

coarse mesh



Courant Number for Coupled Solver

Time step t computed from CFL (Courant-Friedrichs-Lewy) condition:

In general, for coupled/explicit solver: Courant number cannot be > 2.0 from stability constraints; default = 1.0. May have to be set lower for problems with complex physics (e.g., high

Mach number). Defines time step t for time-explicit unsteady solver.

For coupled/implicit solver: Courant number can be set to 10, 100, or even higher Multigrid solver can reduce CFL if divergence is detected

u

xt

)CFL( CFL = Courant numberwhere u = appropriate velocity scalex = grid spacing



Increasing the Courant Number

To accelerate convergence, increase the Courant number if all residuals are either:

Monotonically decreasing Oscillating very slightly, but

decreasing

Solve Controls Solution...



Stability

If residuals are increasing or “stuck,” less aggressive solver settings are needed.

Try one or more of these changes: Decrease under-relaxation for equation(s) having convergence trouble.

(segregated solvers) Reduce Courant number (coupled solver). Adjust multigrid parameters. Try to compute an initial solution with a first-order discretization scheme.



Residual plot showing poor convergence behavior improved by a decrease in under-relaxation:

Example: Improving Stability

Under-relaxation decreased here



Accuracy

Once you have a converged solution, need to be sure it is accurate.

Compute converged solution.

Increase to second-order discretization scheme (if not already), and recalculate a converged solution.

Examine results for significant flow features or variables.

Refine grid and recalculate a converged solution.

Compare results w/results for last grid.

Solution different?Yes No

Solution isgrid-independent!



Choosing the Discretization Scheme

First-order: computes values at faces using values at adjacent cells. Faster convergence; less accurate

Second-order: computes face values using values at adjacent cells and their neighbors.

Slower convergence; more accurate Provides more accurate results, especially for tri/tet meshes. When flow is not aligned with the grid, first-order scheme increases

discretization error. In general, you should try to obtain a second-order solution.

face

adjacent cells



Grid Adaption

In general, finer grid = more accurate solution. Grid adaption allows you to add more cells where they are needed to

resolve the flow field. Example: regions of rapidly-changing velocity or pressure

If converged solution is not accurate enough: Adapt grid in appropriate region(s). Then continue calculation to convergence.

Use solution on original grid as initial solution. Data from original grid is automatically interpolated to finer grid.

Repeat adaption/calculation procedure if necessary. When solution no longer changes with further grid adaption, you have

a “grid-independent” solution.



Example: Supersonic Flow Over a Cylinder

Adapt grid in regions of high pressure gradient because there is a jump in pressure across the shock.

Adapt Gradient...

Original mesh Mesh after pressure gradient adaption



Example: Impinging Jet

Adapt grid in regions of high x velocity.

Adapt Iso-Value...

Original mesh shown withcontours of x velocity

Mesh after x-velocity isovalue adaption



Adaption Guidelines

In general, avoid large volume variations in regions with large gradients.

If you know where gradients are expected (e.g., shock, boundary layer), concentrate the original grid in that region.

If you need to adapt in a region with a large gradient, try to adapt uniformly.



Determining Grid Independence

Compare results obtained w/different grids to determine grid independence:

Overall pressure drop (Report Surface Integrals...) Overall heat transfer coefficient (Report Surface Integrals...) Lift, drag, and moment coefficients (Report Forces...) Average value of a variable at a point or on a surface (Report Surface

Integrals...) Average exit pressure, temperature at a significant location, etc.

Variation of a variable value along a line (Plot XY Plot...) Flow behaving “as expected” (use numerical and graphical reports)



Summary

Solution procedure for the segregated and coupled solvers is the same: Calculate until you get a converged solution. Obtain second-order solution (recommended). Refine grid and recalculate until grid-independent solution is obtained.

All solvers provide tools for judging and improving convergence and ensuring stability.

All solvers provide tools for checking and improving accuracy. Solution accuracy will depend on the appropriateness of the

physical models that you choose and the boundary conditions that you specify.

© Fluent Inc. 11/29/2015E1 Fluent Software Training TRN-98-006 Using the Segregated and Coupled...

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Transcript of © Fluent Inc. 11/29/2015E1 Fluent Software Training TRN-98-006 Using the Segregated and Coupled...